A Bayes Formula for Nonlinear Filtering with Gaussian and Cox Noise

Abstract

A Bayes-type formula is derived for the nonlinear filter where the observation contains both general Gaussian noise as well as Cox noise whose jump intensity depends on the signal. This formula extends the well-known Kallianpur-Striebel formula in the classical non-linear filter setting. We also discuss Zakai-type equations for both the unnormalized conditional distribution as well as unnormalized conditional density in case the signal is a Markovian jump diffusion.

1. Introduction

The general filtering setting can be described as follows. Assume a partially observable process (𝑋,𝑌)=(𝑋_𝑡,𝑌_𝑡)_{0≤𝑡≤𝑇}∈ℝ² defined on a probability space (Ω,ℱ,ℙ). The real valued process 𝑋_𝑡 stands for the unobservable component, referred to as the signal process or system process, whereas 𝑌_𝑡 is the observable part, called observation process. Thus information about 𝑋_𝑡 can only be obtained by extracting the information about 𝑋 that is contained in the observation 𝑌_𝑡 in a best possible way. In filter theory this is done by determining the conditional distribution of 𝑋_𝑡 given the information 𝜎-field ℱ^𝑌_𝑡 generated by 𝑌_𝑠,0≤𝑠≤𝑡. Or stated in an equivalent way, the objective is to compute the optimal filter as the conditional expectation𝐸_ℙ𝑓𝑋_𝑡∣ℱ^𝑌_𝑡 (1.1)

for a rich enough class of functions 𝑓.

In the classical nonlinear filter setting, the dynamics of the observation process 𝑌_𝑡 is supposed to follow the following Itô process 𝑑𝑌_𝑡=ℎ𝑡,𝑋_𝑡𝑑𝑡+𝑑𝑊_𝑡, (1.2)

where 𝑊_𝑡 is a Brownian motion independent of 𝑋. Under certain conditions on the drift ℎ(𝑡,𝑋_𝑡) (see [1, 2]), Kallianpur and Striebel derived a Bayes-type formula for the conditional distribution expressed in terms of the so-called unnormalized conditional distribution. In the special case when the dynamics of the signal follows an Itô diffusion 𝑑𝑋_𝑡=𝑏𝑡,𝑋_𝑡𝑑𝑡+𝜎𝑡,𝑋_𝑡𝑑𝐵_𝑡, (1.3)

for a second Brownian motion 𝐵_𝑡, Zakai [3] showed under certain conditions that the unnormalized conditional density is the solution of an associated stochastic partial differential equation, the so called Zakai equation.

In this paper we extend the classical filter model to the following more general setting. For a general signal process 𝑋 we suppose that the observation model is given as𝑌_𝑡=𝛽(𝑡,𝑋)+𝐺_𝑡+^𝑡_0ℝ0𝜍𝑁_𝜆(𝑑𝑡,𝑑𝜍), (1.4) where (i)𝐺_𝑡 is a general Gaussian process with zero mean and continuous covariance function 𝑅(𝑠,𝑡),0≤𝑠,𝑡≤𝑇, that is, independent of the signal process 𝑋; (ii)Let ℱ^𝑌_𝑡 (resp. ℱ^𝑋_𝑡) denote the 𝜎-algebra generated by {𝑌_𝑠,0≤𝑠≤𝑡} (resp. {𝑋_𝑠,0≤𝑠≤𝑡}) augmented by the null sets. Define the filtration (ℱ_𝑡)_{0≤𝑡≤𝑇} through ℱ_𝑡∶=ℱ^𝑋_𝑇∨ℱ^𝑌_𝑡. Then we assume that the process𝐿_𝑡∶=^𝑡_0ℝ0𝜍𝑁_𝜆(𝑑𝑡,𝑑𝜍) (1.5)is a pure jump ℱ_𝑡-semimartingale determined through the integer-valued random measure 𝑁_𝜆 that has an ℱ_𝑡-predictable compensator of the form 𝜇(𝑑𝑡,𝑑𝜍,𝜔)=𝜆(𝑡,𝑋,𝜍)𝑑𝑡𝜈(𝑑𝜍) (1.6)for a Lévy measure 𝜈 and a functional 𝜆(𝑡,𝑋(𝜔),𝜍). In particular, 𝐺_𝑡 and 𝐿_𝑡 are independent; (iii)the function 𝛽∶[0,𝑇]×ℝ^[0,𝑇]→ℝ is such that 𝛽(𝑡,⋅) is ℱ^𝑋_𝑡-measurable and 𝛽(⋅,𝑋(𝜔)) is in 𝐻(𝑅) for almost all 𝜔, where 𝐻(𝑅) denotes the Hilbert space generated by 𝑅(𝑠,𝑡) (see Section 2).

The observation dynamics consists thus of an information drift of the signal disturbed by some Gaussian noise plus a pure jump part whose jump intensity depends on the signal. Note that a jump process of the form given above is also referred to as Cox process.

The objective of the paper is in a first step to extend the Kallianpur-Striebel Bayes type formula to the generalized filter setting from above. When there are no jumps present in the observation dynamics (1.4), the corresponding formula has been developed in [4]. We will extend their way of reasoning to the situation including Cox noise.

In a second step we then derive a Zakai-type measure valued stochastic differential equations for the unnormalized conditional distribution of the filter. For this purpose we assume the signal process 𝑋 to be a Markov process with generator 𝒪_𝑡∶=ℒ_𝑡+ℬ_𝑡 given as ℒ_𝑡𝑓(𝑥)∶=𝑏(𝑡,𝑥)𝜕_𝑥𝑓(𝑥)+12𝜎²(𝑡,𝑥)𝜕_𝑥𝑥𝑓(𝑥),ℬ_𝑡𝑓(𝑥)∶=_ℝ0𝑓𝑥+𝛾(𝑡,𝑥)𝜍−𝑓(𝑥)−𝜕_𝑥𝑓(𝑥)𝛾(𝑡,𝑥)𝜍𝜐(𝑑𝜍), (1.7)

with the coefficients 𝑏(𝑡,𝑥),𝜎(𝑡,𝑥),and𝛾(𝑡,𝑥) and 𝑓(𝑥) being in 𝐶²₀(ℝ) for every 𝑡. Here, 𝐶²₀(ℝ) is the space of continuous functions with compact support and bounded derivatives up to order 2. Further, we develop a Zakai-type stochastic parabolic integropartial differential equation for the unnormalized conditional density, given it exists. In the case the dynamics of 𝑋 does not contain any jumps and the Gaussian noise 𝐺_𝑡 in the observation is Brownian motion, the corresponding Zakai equation was also studied in [5]. We further refer to [6] where nonlinear filtering for jump diffusions is considered. For further information on Zakai equations in a semimartingale setting we refer to [7, 8].

The remaining part of the paper is organized as follows. in Section 2 we briefly recall some theory of reproducing kernel Hilbert spaces. In Section 3 we obtain the Kallianpur-Striebel formula, before we discuss the Zakai-type equations in Section 4.

2. Reproducing Kernel Hilbert Space and Stochastic Processes

A Hilbert space 𝐻 consisting of real valued functions on some set 𝐓 is said to be a reproducing kernel Hilbert space (RKHS), if there exists a function 𝐾 on 𝐓×𝐓 with the following two properties: for every 𝑡 in 𝐓 and 𝑔 in 𝐻, (i)𝐾(⋅,𝑡)∈𝐻, (ii)(𝑔(⋅),𝐾(⋅,𝑡))=𝑔(𝑡). (The reproducing property)

𝐾 is called the reproducing kernel of 𝐻. The following basic properties can be found in [9]. (1)If a reproducing kernel exists, then it is unique. (2)If 𝐾 is the reproducing kernel of a Hilbert space 𝐻, then {𝐾(⋅,𝑡),𝑡∈𝐓} spans 𝐻. (3)If 𝐾 is the reproducing kernel of a Hilbert space 𝐻, then it is nonnegative definite in the sense that for all 𝑡₁,…,𝑡_𝑛 in 𝐓 and 𝑎₁,…,𝑎_𝑛∈ℝ^𝑛_𝑖,𝑗=1𝐾𝑡_𝑖,𝑡_𝑗𝑎_𝑖𝑎_𝑗≥0. (2.1)

The converse of (3), stated in Theorem 2.1 below, is fundamental towards understanding the RKHS representation of Gaussian processes. A proof of the theorem can be found in [9].

Theorem 2.1 (E. H. Moore). A symmetric nonnegative definite function 𝐾 on 𝐓×𝐓 generates a unique Hilbert space, which we denote by 𝐻(𝐾) or sometimes by 𝐻(𝐾,𝐓), of which 𝐾 is the reproducing kernel.

Now suppose 𝐾(𝑠,𝑡),𝑠,𝑡∈𝐓, is a nonnegative definite function. Then, by Theorem 2.1, there is a RKHS, 𝐻(𝐾,𝐓), with 𝐾 as its reproducing kernel. If we restrict 𝐾 to 𝐓′×𝐓′ where 𝐓′⊂𝐓, then 𝐾 is still a nonnegative definite function. Hence 𝐾 restricted to 𝐓′×𝐓′ will also correspond to a reproducing kernel Hilbert space 𝐻(𝐾,𝐓′) of functions defined on 𝐓′. The following result from ([9, pp. 351]) explains the relationship between these two.

Theorem 2.2. Suppose 𝐾_𝐓, defined on 𝐓×𝐓, is the reproducing kernel of the Hilbert space 𝐻(𝐾_𝐓) with the norm ‖⋅‖. Let 𝐓′⊂𝐓 and 𝐾_𝐓′ be the restriction of 𝐾_𝐓 on 𝐓′×𝐓′. Then 𝐻(𝐾_𝐓′) consists of all 𝑓 in H(𝐾_𝐓) restricted to 𝐓′. Further, for such a restriction 𝑓′∈𝐻(𝐾_𝐓′) the norm ‖𝑓′‖_{𝐻(𝐾𝐓′)} is the minimum of ‖𝑓‖_{𝐻(𝐾𝐓)} for all 𝑓∈𝐻(𝐾_𝐓) whose restriction to 𝐓′ is 𝑓′.

If 𝐾(𝑠,𝑡) is the covariance function for some zero mean process 𝑍_𝑡,𝑡∈𝐓, then, by Theorem 2.1, there exists a unique RKHS, 𝐻(𝐾,𝐓), for which 𝐾 is the reproducing kernel. It is also easy to see (e.g., see [10, Theorem 3D]) that there exists a congruence (linear, one-to-one, inner product preserving map) between 𝐻(𝐾) and sp^𝐿2{𝑍_𝑡,𝑡∈𝐓} which takes 𝐾(⋅,𝑡) to 𝑍_𝑡. Let us denote by ⟨𝑍,ℎ⟩∈sp^𝐿2{𝑍_𝑡,𝑡∈𝐓}, the image of ℎ∈𝐻(𝐾,𝐓) under the congruence.

We conclude the section with an important special case.

2.1. A Useful Example

Suppose the stochastic process 𝑍_𝑡 is a Gaussian process given by 𝑍_𝑡=^𝑡₀𝐹(𝑡,𝑢)𝑑𝑊_𝑢,0≤𝑡≤𝑇, (2.2)

where ∫^𝑡₀𝐹²(𝑡,𝑢)𝑑𝑢<∞ for all 0≤𝑡≤𝑇 and 𝑊_𝑢 is Brownian motion. Then the covariance function 𝐾(𝑠,𝑡)≡𝐸𝑍_𝑠𝑍_𝑡=₀^𝑡∧𝑠𝐹(𝑡,𝑢)𝐹(𝑠,𝑢)𝑑𝑢, (2.3) and the corresponding RKHS is given by 𝐻(𝐾)=𝑔∶𝑔(𝑡)=^𝑡₀𝐹(𝑡,𝑢)𝑔^∗(𝑢)𝑑𝑢,0≤𝑡≤𝑇 (2.4) for some (necessarily unique) 𝑔^∗∈sp^𝐿2{𝐹(𝑡,⋅)1_[0,𝑡](⋅),0≤𝑡≤𝑇}, with the inner product𝑔₁,𝑔_2𝐻(𝐾)=^𝑇₀𝑔^∗₁(𝑢)𝑔^∗₂(𝑢)𝑑𝑢, (2.5)

where𝑔₁(𝑠)=^𝑠₀𝐹(𝑠,𝑢)𝑔^∗₁(𝑢)𝑑𝑢,𝑔₂(𝑠)=^𝑠₀𝐹(𝑠,𝑢)𝑔^∗₂(𝑢)𝑑𝑢. (2.6)

For 0≤𝑡≤𝑇, by taking 𝐾(⋅,𝑡)^∗ to be 𝐹(𝑡,⋅)1_[0,𝑡](⋅), we see, from (2.3) and (2.4), that 𝐾(⋅,𝑡)∈𝐻(𝐾). To check the reproducing property suppose ℎ(𝑡)=∫^𝑡₀𝐹(𝑡,𝑢)ℎ^∗(𝑢)𝑑𝑢∈𝐻(𝐾). Then(ℎ,𝐾(⋅,𝑡))_𝐻(𝐾)=^𝑇₀ℎ^∗(𝑢)𝐾(⋅,𝑡)^∗𝑑𝑢=^𝑡₀ℎ^∗(𝑢)𝐹(𝑡,𝑢)𝑑𝑢=ℎ(𝑡). (2.7)

Also, in this case, it is very easy to check (cf. [11], Theorem 4D) that the congruence between 𝐻(𝐾) and sp^𝐿2{𝑍_𝑡,𝑡∈𝐓} is given by 𝑍,𝑔=^𝑇₀𝑔^∗(𝑢)𝑑𝑊_𝑢. (2.8)

3. The Filter Setting and a Bayes Formula

Assume a partially observable process (𝑋,𝑌)=(𝑋_𝑡,𝑌_𝑡)_{0≤𝑡≤𝑇}∈ℝ² defined on a probability space (Ω,ℱ,ℙ). The real valued process 𝑋_𝑡 stands for the unobservable component, referred to as the signal process, whereas 𝑌_𝑡 is the observable part, called observation process. In particular, we assume that the dynamics of the observation process is given as follows:𝑌_𝑡=𝛽(𝑡,𝑋)+𝐺_𝑡+^𝑡_0ℝ0𝜍𝑁_𝜆(𝑑𝑡,𝑑𝜍), (3.1) where (i)𝐺_𝑡 is a Gaussian process with zero mean and continuous covariance function 𝑅(𝑠,𝑡),0≤𝑠,𝑡≤𝑇, that is, independent of the signal process 𝑋; (ii)the function 𝛽∶[0,𝑇]×ℝ^[0,𝑇]→ℝ is such that 𝛽(𝑡,⋅) is ℱ^𝑋_𝑡-measurable and 𝛽(⋅,𝑋(𝜔)) is in 𝐻(𝑅) for almost all 𝜔, where 𝐻(𝑅) denotes the Hilbert space generated by 𝑅(𝑠,𝑡) (see Section 2); (iii)let ℱ^𝑌_𝑡 (resp., ℱ^𝑋_𝑡) denote the 𝜎-algebra generated by {𝑌_𝑠,0≤𝑠≤𝑡} (resp., {𝑋_𝑠,0≤𝑠≤𝑡}) augmented by the null sets. Define the filtration (ℱ_𝑡)_{0≤𝑡≤𝑇} through ℱ_𝑡∶=ℱ^𝑋_𝑇∨ℱ^𝑌_𝑡. Then we assume that the process𝐿_𝑡∶=^𝑡_0ℝ0𝜍𝑁_𝜆(𝑑𝑡,𝑑𝜍) (3.2)is a pure jump ℱ_𝑡-semimartingale determined through the integer-valued random measure 𝑁_𝜆 that has an ℱ_𝑡-predictable compensator of the form 𝜇(𝑑𝑡,𝑑𝜍,𝜔)=𝜆(𝑡,𝑋,𝜍)𝑑𝑡𝜈(𝑑𝜍) (3.3)for a Lévy measure 𝜈 and a functional 𝜆(𝑡,𝑋(𝜔),𝜍); (iv)the functional 𝜆(𝑡,𝑋,𝜍) is assumed to be strictly positive and such that^𝑇_0ℝ0log²(𝜆(𝑠,𝑋,𝜍))𝜇(𝑑𝑠,𝑑𝜍)<∞a.s.,_𝑇0ℝ0log²(𝜆(𝑠,𝑋,𝜍))𝑑𝑠𝜈(𝑑𝜍)<∞a.s., (3.4)Λ_𝑡∶=exp^𝑡_0ℝ01log𝜆()𝑁𝑠,𝑋,𝜍^𝜆()+𝑑𝑠,𝑑𝜍_𝑡0ℝ01log𝜆()−1𝑠,𝑋,𝜍𝜆()𝜇(𝑠,𝑋,𝜍+1𝑑𝑠,𝑑𝜍) (3.5)

is a well-defined ℱ_𝑡-martingale. Here 𝑁_𝜆(𝑑𝑠,𝑑𝜍) stands for the compensated jump measure𝑁_𝜆(𝑑𝑠,𝑑𝜍)∶=𝑁_𝜆(𝑑𝑠,𝑑𝜍)−𝜇(𝑑𝑡,𝑑𝜍). (3.6)

Remark 3.1. Note that the specific from of the predictable compensator 𝜇(𝑑𝑡,𝑑𝜍,𝜔)  implies that  𝐿_𝑡  is a process with conditionally independent increments with respect to the  𝜎-algebra  ℱ^𝑋_𝑇, that is, 𝐸_ℙ𝑓𝐿_𝑡−𝐿_𝑠1_𝐴∣ℱ^𝑋_𝑇=𝐸_ℙ𝑓𝐿_𝑡−𝐿_𝑠∣ℱ^𝑋_𝑇𝐸_ℙ1_𝐴∣ℱ^𝑋_𝑇, (3.7) for all bounded measurable functions 𝑓, 𝐴∈ℱ_𝑠, and 0≤𝑠<𝑡≤𝑇 (see, e.g., in [12, Theorem 6.6]). Also, it follows that the processes 𝐺 is independent of the random measure 𝑁_𝜆(𝑑𝑠,𝑑𝜍).

Given a Borel measurable function 𝑓, our nonlinear filtering problem then comes down to determine the least square estimate of 𝑓(𝑋_𝑡), given the observations up to time 𝑡. In other words, the problem consists in evaluating the optimal filter 𝐸_ℙ𝑓𝑋_𝑡∣ℱ^𝑌_𝑡. (3.8) In this section we want to derive a Bayes formula for the optimal filter (3.8) by an extension of the reference measure method presented in [4] for the purely Gaussian case. For this purpose, define for each 0≤𝑡≤𝑇 with 𝛽(⋅)=𝛽(⋅,𝑋)Λ′_𝑡∶=exp−𝐺,𝛽_𝑡−12𝛽²_𝑡. (3.9)

Then the main tool is the following extension of Theorem 3.1 in [4].

Lemma 3.2. Define 𝑑ℚ∶=Λ_𝑡Λ′_𝑡𝑑ℙ. (3.10) Then ℚ_𝑡 is a probability measure, and under ℚ_𝑡 we have that 𝑌_𝑡=𝐺_𝑡+𝐿_𝑡, (3.11) where 𝐺_𝑠=𝛽(𝑠,𝑋)+𝐺_𝑠, 0≤𝑠≤𝑡, is a Gaussian process with zero mean and covariance function 𝑅, 𝐿_𝑠, 0≤𝑠≤𝑡, is a pure jump Lévy process with Lévy measure 𝜈, and the process 𝑋_𝑠, 0≤𝑠≤𝑇 has the same distribution as under ℙ. Further, the processes 𝐺, 𝐿, and 𝑋 are independent under ℚ_𝑡.

Proof. Fix 0≤𝑡≤𝑇. First note that since 𝛽(⋅)∈𝐻(𝑅) almost surely, we have by Theorem 2.2 that 𝛽|_[0,𝑡]∈𝐻(𝑅;𝑡) almost surely. Further, by the independence of the Gaussian process 𝐺 from 𝑋 and from the random measure 𝑁_𝜆(𝑑𝑠,𝑑𝜍) it follows that 𝐸_ℙΛ_𝑡Λ^′_𝑡=𝐸_ℙ𝐸_ℙΛ_𝑡∣ℱ^𝑋_𝑇𝐸_ℙΛ^′_𝑡∣ℱ^𝑋_𝑇. (3.12) Since for 𝑓∈𝐻(𝑅;𝑡) the random variable ⟨𝐺,𝑓⟩_𝑡 is Gaussian with zero mean and variance ‖𝑓‖²_𝑡, it follows again by the independence of 𝐺 from 𝑋 and the martingale property of Λ_𝑡 that 𝐸_ℙ[Λ_𝑡Λ′_𝑡]=1, and ℚ_𝑡 is a probability measure.
Now take 0≤𝑠₁,…,𝑠_𝑚≤𝑡,0≤𝑟₁,…,𝑟_𝑝≤𝑡,0≤𝑡₁,…,𝑡_𝑛≤𝑇 and real numbers 𝜆₁,…,𝜆_𝑚, 𝛾₁,…,𝛾_𝑝, 𝛼₁,…,𝛼_𝑛 and consider the joint characteristic function 𝐸_ℚ𝑡𝑒^{𝑖∑𝑛𝑗=1𝛼𝑗𝑋𝑡𝑗+𝑖∑𝑚𝑖=1𝜆𝑖}𝐺_{𝑠𝑖+𝑖∑}^𝑝_{𝑘=1𝛾𝑘(𝐿𝑟𝑘−𝐿𝑟k−1)}=𝐸_ℙ𝑒^{𝑖∑𝑛𝑗=1𝛼𝑗𝑋𝑡𝑗+𝑖∑𝑚𝑖=1𝜆𝑖}𝐺_{𝑠𝑖+𝑖∑}^𝑝_{𝑘=1𝛾𝑘(𝐿𝑟𝑘−𝐿𝑟𝑘−1)}Λ_𝑡Λ^′_𝑡=𝐸_ℙ𝑒^{𝑖∑𝑛𝑗=1𝛼𝑗𝑋𝑡𝑗}𝐸_ℙ𝑒^{𝑖∑𝑚𝑖=1𝜆𝑖}𝐺_𝑠𝑖Λ^′_𝑡∣ℱ^𝑋_𝑇𝐸_ℙ𝑒^{𝑖∑𝑝𝑘=1𝛾𝑘(𝐿𝑟𝑘−𝐿𝑟𝑘−1)}Λ_𝑡∣ℱ^𝑋_𝑇. (3.13) Here, for computational convenience, the part of the characteristic function that concerns 𝐿 is formulated in terms of increments of 𝐿 (where we set 𝑟₀=0). Now, as in in [4, Theorem 3.1], we get by the independence of 𝐺 from 𝑋 that 𝐸_ℙ𝑒^{𝑖∑𝑚𝑖=1𝜆𝑖}𝐺_𝑠𝑖Λ′_𝑡∣ℱ^𝑋_𝑇=𝑒^{−∑𝑚𝑖,𝑙=1𝜆𝑖𝜆𝑙𝑅(𝑠𝑖,𝑠𝑙)}, (3.14) which is the characteristic function of a Gaussian process with mean zero and covariance function 𝑅.
Further, by the conditional independent increments of 𝐿 we get like in the proof of in [12, Theorem 6.6] that 𝐸_ℙ𝑒^{∫𝑢𝑟∫ℝ0𝛿(𝑠,𝑋,𝜍)}𝑁_{𝜆(𝑑𝑠,𝑑𝜍)}∣ℱ^𝑋_𝑇=𝑒^{∫𝑢𝑟∫ℝ0(𝑒𝛿(𝑠,𝑋,𝜍)−1−𝛿(𝑠,𝑋,𝜍))𝜇(𝑑𝑡,𝑑𝜍)} (3.15) for 0≤𝑟≤𝑢≤𝑇. So that for one increment one obtains 𝐸_ℙ𝑒^{𝑖𝛾(𝐿𝑢−𝐿r)}Λ_𝑡∣ℱ^𝑋_𝑇=𝐸_ℙexp^𝑢_𝑟ℝ01𝑖𝛾𝜍+log𝜆()𝑁𝑠,𝑋,𝜍^𝜆()+𝑑𝑠,𝑑𝜍_𝑢𝑟ℝ01𝑖𝛾𝜍+log𝜆()−1𝑠,𝑋,𝜍𝜆()𝜇(𝑠,𝑋,𝜍+1𝑑𝑡,𝑑𝜍)∣ℱ^𝑋_𝑇=𝐸_ℙexp^𝑢_𝑟ℝ0𝑒^{𝑖𝛾𝜍+log(1/𝜆(𝑠,𝑋,𝜍))}1−1−𝑖𝛾𝜍−log𝜆()𝜇()+𝑠,𝑋,𝜍𝑑𝑡,𝑑𝜍_𝑢𝑟ℝ01𝑖𝛾𝜍+log𝜆()−1𝑠,𝑋,𝜍𝜆()𝜇(𝑠,𝑋,𝜍+1𝑑𝑡,𝑑𝜍)∣ℱ^𝑋_𝑇=𝐸_ℙexp^𝑢_𝑟ℝ0𝑒^{𝑖𝛾𝜍+log(1/𝜆(𝑠,𝑋,𝜍))}−1𝜆()𝜆()(𝑠,𝑋,𝜍𝑡,𝑋,𝜍𝑑𝑡𝜈𝑑𝜍)∣ℱ^𝑋_𝑇=exp(𝑢−𝑟)_ℝ0𝑒^𝑖𝛾𝜍−1𝜈(𝑑𝜍). (3.16) The generalization to the sum of increments is straightforward, and one obtains the characteristic function of the finite dimensional distribution of a Lévy process (of finite variation): 𝐸_ℙ𝑒^{𝑖∑𝑝𝑘=1𝛾𝑘(𝐿𝑟𝑘−𝐿𝑟𝑘−1)}Λ_𝑡∣ℱ^𝑋_𝑇=exp^𝑝_𝑘=1𝑟_𝑘−𝑟_𝑘−1ℝ0𝑒^{𝑖𝛾𝑘𝜍}−1𝜈(𝑑𝜍). (3.17) All together we end up with 𝐸_ℚ𝑡𝑒^{𝑖∑𝑛𝑗=1𝛼𝑗𝑋𝑡𝑗+𝑖∑𝑚𝑖=1𝜆𝑖}𝐺_{𝑠𝑖+𝑖∑}^𝑝_{𝑘=1𝛾𝑘(𝐿𝑟𝑘−𝐿𝑟𝑘−1)}=𝐸_ℙ𝑒^{𝑖∑𝑛𝑗=1𝛼𝑗𝑋𝑡𝑗}⋅𝑒^{−∑𝑚𝑖,𝑙=1𝜆𝑖𝜆𝑙𝑅(𝑠𝑖,𝑠𝑙)}⋅𝑒^{∑𝑝𝑘=1(𝑟𝑘−𝑟𝑘−1)∫ℝ0(𝑒𝑘𝜍𝑖𝛾−1)𝜈(𝑑𝜍)}, (3.18) which completes the proof.

Remark 3.3. Note that in case  𝐺  is Brownian motion Lemma 3.2 is just the usual Girsanov theorem for Brownian motion and random measures. In this case, it follows from Cameron-Martin's result and the fact that  𝑋  is independent of 𝐺 that  Λ_𝑡Λ^′_𝑡  is a martingale and  𝑑ℚ  is a probability measure.

Now, the inverse Radon-Nikodym derivative𝑑ℙ𝑑ℚ_𝑡=Λ_𝑡⁻¹Λ^′_𝑡⁻¹ (3.19)

is ℚ_𝑡-a.s. by condition (3.4) and an argument like in [4, p. 857] given through Λ_𝑡⁻¹=exp^𝑡_0ℝ0(𝜆(𝑁()+log𝑠,𝑋,𝜍))𝑑𝑠,𝑑𝜍_𝑡0ℝ0(𝜆(()(,Λlog𝑠,𝑋,𝜍))−𝜆𝑠,𝑋,𝜍+1𝑑𝑠𝜈𝑑𝜍)_′𝑡−1=exp𝐺,𝛽_𝑡−12𝛽²_𝑡. (3.20)

Here𝑁(𝑑𝑠,𝑑𝜍)∶=𝑁_𝜆(𝑑𝑠,𝑑𝜍)−𝑑𝑡𝜈(𝑑𝜍) (3.21)

is now a compensated Poisson random measure under ℚ_𝑡. Then we have by the Bayes formula for conditional expectation for any ℱ^𝑋_𝑇-measurable integrable function 𝑔(𝑇,𝑋)𝐸_ℙ𝑔(𝑇,𝑋)∣ℱ^𝑌_𝑡=𝐸_ℚ𝑡𝑔(𝑇,𝑋)Λ_𝑡⁻¹Λ^′_𝑡⁻¹∣ℱ^𝑌_𝑡𝐸_ℚ𝑡Λ_𝑡⁻¹Λ^′_𝑡⁻¹∣ℱ^𝑌_𝑡. (3.22)

From Lemma 3.2 we know that the processes (𝐺_𝑠)_{0≤𝑠≤𝑡}, (𝐿_𝑠)_{0≤𝑠≤𝑡}, and (𝑋_𝑠)_{0≤𝑠≤𝑇} are independent under ℚ_𝑡 and that the distribution of 𝑋 is the same under ℚ_𝑡 as under ℙ. Hence conditional expectations of the form 𝐸_ℚ𝑡[𝜙(𝑋,𝐺,𝐿)∣ℱ^𝑌_𝑡] can be computed as 𝐸_ℚ𝑡𝜙𝑋,𝐺,𝐿∣ℱ^𝑌_𝑡(𝜔)=_Ω𝜙𝑋(𝜔),𝐺(𝜔),𝐿(𝜔)ℚ_𝑡(𝑑𝜔)=_Ω𝜙𝑋(𝜔),𝐺(𝜔)(𝜔ℙ(𝑑𝜔)ℙ,𝐿)=𝐸𝜙𝑋(𝜔),𝐺(𝜔),𝐿(𝜔), (3.23)

where (𝜔,𝜔)∈Ω×Ω and the index ℙ denotes integration with respect to 𝜔. Consequently, we get the following Bayes formula for the optimal filter.

Theorem 3.4. Under the above specified conditions, for any ℱ^𝑋_𝑇-measurable integrable function 𝑔(𝑇,𝑋)𝐸_ℙ𝑔(𝑇,𝑋)∣ℱ^𝑌_𝑡=∫_Ω𝑔(𝑇,𝑋(𝜔))𝛼_𝑡(𝜔,𝜔)𝛼^′_𝑡(𝜔,𝜔)ℙ(𝑑𝜔)∫_Ω𝛼_𝑡(𝜔,𝜔)𝛼^′_𝑡(𝜔)ℙ(𝑑𝜔)=𝐸ℙ𝑔((𝜔𝛼𝜔,𝑇,𝑋))_𝑡(𝜔,𝜔)𝛼^′_𝑡(𝜔,𝜔)𝐸ℙ𝛼_𝑡(𝜔,𝜔)𝛼^′_𝑡(𝜔,𝜔,) (3.24) where 𝛼_𝑡(𝜔,𝜔)=exp^𝑡_0ℝ0(𝜆((𝜔)𝑁()+log𝑠,𝑋,𝜍))𝜔,𝑑𝑠,𝑑𝜍_𝑡0ℝ0(𝜆((𝜔)((𝜔))(,𝛼log𝑠,𝑋,𝜍))−𝜆𝑠,𝑋,𝜍+1𝑑𝑠𝜈𝑑𝜍)_′𝑡(𝜔,𝜔)=exp𝐺(𝜔),𝛽(⋅,𝜔)_𝑡−12𝛽(⋅,𝜔)²_𝑡. (3.25)

4. Zakai-Type Equations

Using the Bayes formula from above we now want to proceed further in deriving a Zakai-type equations for the unnormalized filter. This equation is basic in order to obtain the filter recursively. To this end we have to impose certain restrictions on both the signal process and the Gaussian part of the observation process.

Regarding the signal process 𝑋, we assume its dynamics to be Markov. More precisely, we consider the parabolic integrodifferential operator 𝒪_𝑡∶=ℒ_𝑡+ℬ_𝑡, where ℒ_𝑡𝑓(𝑥)∶=𝑏(𝑡,𝑥)𝜕_𝑥𝑓(𝑥)+12𝜎²(𝑡,𝑥)𝜕_𝑥𝑥𝑓(𝑥),ℬ_𝑡𝑓(𝑥)∶=_ℝ0𝑓𝑥+𝛾(𝑡,𝑥)𝜍−𝑓(𝑥)−𝜕_𝑥𝑓(𝑥)𝛾(𝑡,𝑥)𝜍𝜐(𝑑𝜍), (4.1)

for 𝑓∈𝐶²₀(ℝ). Here, 𝐶²₀(ℝ) is the space of continuous functions with compact support and bounded derivatives up to order 2. Further, we suppose that 𝑏(𝑡,⋅), 𝜎(𝑡,⋅), and 𝛾(𝑡,⋅) are in 𝐶²₀(ℝ) for every 𝑡 and that 𝜐(𝑑𝜍) is a Lévy measure with second moment. The signal process 𝑋_𝑡,0≤𝑡≤𝑇, is then assumed to be a solution of the martingale problem corresponding to 𝒪_𝑡, that is,𝑓𝑋_𝑡−^𝑡₀𝒪_𝑢𝑓𝑋_𝑢𝑑𝑢 (4.2)

is a ℱ^𝑋_𝑡-martingale with respect to ℙ for every 𝑓∈𝐶²₀(ℝ).

Further, we restrict the Gaussian process 𝐺 of the observation process in (3.1) to belong to the special case presented in Section 2.1, that is,𝐺_𝑡=^𝑡₀𝐹(𝑡,𝑠)𝑑𝑊_𝑠, (4.3)

where 𝑊_𝑡 is Brownian motion and 𝐹(𝑡,𝑠) is a deterministic function such that ∫^𝑡₀𝐹²(𝑡,𝑠)𝑑𝑠, 0≤𝑡≤𝑇. Note that this type of processes both includes Ornstein-Uhlenbeck processes as well as fractional Brownian motion. Then 𝛽(𝑡,𝑋) will be of the form𝛽(𝑡,𝑋)=^𝑡₀𝐹(𝑡,𝑠)ℎ𝑠,𝑋_𝑠𝑑𝑠. (4.4)

Further, with𝑊_𝑡∶=^𝑡₀ℎ𝑠,𝑋_𝑠𝑑𝑠+𝑊_𝑡 (4.5)

we get ⟨𝐺,𝛽⟩_𝑡=∫^𝑡₀ℎ(𝑠,𝑋_𝑠)𝑑𝑊_𝑠 and ‖𝛽‖²_𝑡=∫^𝑡₀ℎ²(𝑠,𝑋_𝑠)𝑑𝑠, and 𝛼^′_𝑡(𝜔,𝜔) in Theorem 3.4 becomes𝛼^′_𝑡(𝜔,𝜔)=exp^𝑡₀ℎ𝑠,𝑋_𝑠(𝜔)𝑑𝑊_𝑠(𝜔)−12^𝑡₀ℎ²𝑠,𝑋_𝑠(𝜔)𝑑s. (4.6)

Note that in this case 𝑊_𝑠,0≤𝑠≤𝑡, is a Brownian motion under ℚ_𝑡.

For 𝑓∈𝐶²₀(ℝ) we now define the unnormalized filter 𝑉_𝑡(𝑓)=𝑉_𝑡(𝑓)(𝜔) by𝑉_𝑡𝑓(𝜔)∶=_Ω𝑓𝑋_𝑡(𝜔)𝛼_𝑡(𝜔,𝜔)𝛼^′_𝑡(𝜔)ℙ(𝑑𝜔)ℙ𝜔,=𝐸𝑓𝑋_𝑡(𝜔)𝛼_𝑡(𝜔,𝜔)𝛼^′_𝑡(𝜔,𝜔). (4.7)

Then this unnormalized filter obeys the following dynamics.

Theorem 4.1 (Zakai equation I). Under the above specified assumptions, the unnormalized filter 𝑉_𝑡(𝑓) satisfies the equation 𝑑𝑉_𝑡𝑓(⋅)(𝜔)=𝑉_𝑡𝒪_𝑡𝑓(⋅)(𝜔)𝑑𝑡+𝑉_𝑡ℎ(𝑡,⋅)𝑓(⋅)(𝜔)𝑑𝑊_𝑡(𝜔)+_ℝ0𝑉_𝑡(𝜆(𝑡,⋅,𝜍)−1)𝑓(⋅)(𝜔)𝑁(𝜔,𝑑𝑡,𝑑𝜍). (4.8)

Proof. Set 𝑔_𝑡(𝜔)∶=𝑓𝑋_𝑇(𝜔)−^𝑇_𝑡𝒪_𝑠𝑓𝑋_𝑠(𝜔)𝑑𝑠. (4.9) Then, by our assumptions on the coefficients 𝑏, 𝜎, 𝛾 and on the Lévy measure 𝜐(𝑑𝜍), we have |𝑔_𝑡|<𝐶 for some constant 𝐶. Since 𝑓(𝑋_𝑡)−∫^𝑡₀𝒪_𝑡𝑓(𝑋_𝑠)𝑑𝑠 is a martingale we obtain 𝐸ℙ𝑔_𝑡∣ℱ^𝑋(𝜔_)𝑡=𝑓𝑋_𝑡,0≤𝑡≤𝑇. (4.10) If we denote Γ_𝑡(𝜔,𝜔)∶=𝛼_𝑡(𝜔,𝜔)𝛼^′_𝑡(𝜔,𝜔), (4.11) then, because Γ_𝑡(𝜔,𝜔) is ℱ^𝑋(𝜔_)𝑡-measurable for each 𝜔, (4.10) implies that 𝑉_𝑡𝑓ℙ=𝐸𝑓𝑋_𝑡(𝜔)Γ_𝑡(𝜔ℙ𝐸ℙ𝑔𝜔,)=𝐸_𝑡(𝜔)Γ_𝑡(𝜔,𝜔)∣ℱ^𝑋(𝜔_)𝑡ℙ=𝐸𝑔_𝑡(𝜔)Γ_𝑡(𝜔,𝜔). (4.12) By definition of 𝑔_𝑡, 𝑑𝑔_𝑡(𝜔)=𝒪_𝑡𝑓𝑋_𝑡(𝜔)𝑑𝑡. (4.13) Also, Γ_𝑡=Γ_𝑡(𝜔,𝜔) is the Doléans-Dade solution of the following linear SDE: 𝑑Γ_𝑡=ℎ𝑡,𝑋_𝑡(𝜔)Γ_𝑡𝑑𝑊_𝑡(𝜔)+_ℝ0𝜆𝑡,𝑋_𝑡(𝜔),𝜍−1Γ_𝑡𝑁(𝜔,𝑑𝑡,𝑑𝜍). (4.14) So we get 𝐸ℙ𝑔_𝑡(𝜔)Γ_𝑡ℙ=𝐸𝑔₀(𝜔)Γ₀ℙ+𝐸^𝑡₀𝒪_𝑠𝑓𝑋_𝑠(𝜔)Γ_𝑠ℙ𝑑𝑠+𝐸^𝑡₀ℎ𝑠,𝑋_𝑠(𝜔)𝑔_𝑠(𝜔)Γ_𝑠𝑑𝑊_𝑠(𝜔ℙ)+𝐸^𝑡_0ℝ0𝜆𝑠,𝑋_𝑠(𝜔),𝜍−1𝑔_𝑠(𝜔)Γ_𝑠𝑁(𝜔,𝑑𝑠,𝑑𝜍). (4.15) The first term on the right hand side equals 𝑓(𝑋₀), and for the second one we can invoke Fubini's theorem to get 𝐸ℙ^𝑡₀𝒪_𝑠𝑓𝑋_𝑠(𝜔)Γ_𝑠𝑑𝑠=^𝑡₀𝐸ℙ𝒪_𝑠𝑓𝑋_𝑠(𝜔)Γ_𝑠𝑑𝑠=^𝑡₀𝑉_𝑠𝒪_𝑠𝑓(⋅)(𝜔)𝑑s. (4.16) For the third term we employ the stochastic Fubini theorem for Brownian motion (see, for Example 5.14 in [13]) in order to get 𝐸ℙ^𝑡₀ℎ𝑠,𝑋_𝑠(𝜔)𝑔_𝑠(𝜔)Γ_𝑠𝑑𝑊_𝑠(𝜔=_𝑡0)𝐸ℙℎ𝑠,𝑋_𝑠(𝜔)𝑔_𝑠(𝜔)Γ_𝑠𝑑𝑊_𝑠(𝜔)=^𝑡₀𝐸ℙℎ𝑠,𝑋_𝑠(𝜔)Γ_𝑠𝐸ℙ𝑔_𝑠(𝜔)∣ℱ^𝑋(𝜔_)𝑠𝑑𝑊_𝑠(𝜔)=^𝑡₀𝐸ℙℎ𝑠,𝑋_𝑠(𝜔)Γ_𝑠𝑓𝑋_𝑠(𝜔)𝑑𝑊_𝑠(𝜔)=^𝑡₀𝑉_𝑠ℎ(𝑠,⋅)𝑓(⋅)(𝜔)𝑑𝑊_𝑠(𝜔). (4.17) Further, one easily sees that the analogue stochastic Fubini theorem for compensated Poisson random measures holds, and we get analogously for the last term 𝐸ℙ^𝑡_0ℝ0𝜆𝑠,𝑋_𝑠(𝜔),𝜍−1𝑔_𝑠(𝜔)Γ_𝑠𝑁(=𝜔,𝑑𝑠,𝑑𝜍)_𝑡0ℝ0𝑉_𝑠(𝜆(𝑠,⋅,𝜍)−1)𝑓(⋅)(𝜔)𝑁(𝜔,𝑑𝑠,𝑑𝜍), (4.18) which completes the proof.

If one further assumes that the filter has a so-called unnormalized conditional density 𝑢(𝑡,𝑥) then we can derive a stochastic integro-PDE determining 𝑢(𝑡,𝑥) which for the Brownian motion case was first established in [3] and usually is referred to as Zakai equation.

Definition 4.2. We say that a process 𝑢(𝑡,𝑥)=𝑢(𝜔,𝑡,𝑥) is the unnormalized conditional density of the filter if 𝑉_𝑡𝑓(𝜔)=_ℝ𝑓(𝑥)𝑢(𝜔,𝑡,𝑥)𝑑𝑥 (4.19) for all bounded continuous functions 𝑓∶ℝ→ℝ.

From now on we restrict the intergo part ℬ_𝑡 of the operator 𝒪_𝑡 to be the one of a pure jump Lévy process, that is, 𝛾=1, and we assume the initial value 𝑋₀(𝜔) of the signal process to possess a density denoted by 𝜉(𝑥). Then the following holds.

Theorem 4.3 (Zakai equation II). Suppose the unnormalized conditional density 𝑢(𝑡,𝑥) of our filter exists. Then, provided that a solution exists, 𝑢(𝑡,𝑥) solves the following stochastic integro-PDE: 𝑑𝑢(𝑡,𝑥)=𝒪^∗_𝑡𝑢(𝑡,𝑥)𝑑𝑡+ℎ(𝑡,𝑥)𝑢(𝑡,𝑥)𝑑𝑊_𝑡(𝜔)+_ℝ0(𝜆())𝑢()𝑁(),𝑢()(𝑥).𝑡,𝑥,𝜍−1𝑡,𝑥𝜔,𝑑𝑡,𝑑𝜍0,𝑥=𝜉 (4.20) Here 𝒪^∗_𝑡∶=ℒ^∗_𝑡+ℬ^∗_𝑡 is the adjoint operator of 𝒪_𝑡 given through ℒ^∗_𝑡𝑓(𝑥)∶=−𝜕_𝑥𝑏()𝑓(𝑥+1𝑡,𝑥)2𝜕_𝑥𝑥𝜎²()𝑓(𝑥,ℬ𝑡,𝑥)_∗𝑡𝑓(𝑥)∶=_ℝ0𝑓(𝑥−𝜍)−𝑓(𝑥)+𝜕_𝑥𝑓(𝑥)𝜍𝜐(𝑑𝜍), (4.21) for 𝑓∈𝐶²₀(ℝ).

For sufficient conditions on the coefficients under which there exists a classical solution of (4.20) see, for example, [5]; in [14] the existence of solutions in a generalized sense of stochastic distributions is treated.

Proof. By (4.8) and (4.19) we have for all 𝑓∈𝐶^∞₀(ℝ)_ℝ𝑓(𝑥)𝑢(𝑡,𝑥)𝑑𝑥=_ℝ𝑓(𝑥)𝜉(𝑥)𝑑𝑥+^𝑡_0ℝ𝑢(𝑠,𝑥)𝒪_𝑠𝑓(𝑥)+𝑑𝑥𝑑𝑠_𝑡0ℝ𝑢(𝑠,𝑥)ℎ(𝑠,𝑥)𝑓(𝑥)𝑑𝑥𝑑𝑊_𝑠(𝜔)+^𝑡_0ℝ0ℝ𝑢(𝑠,𝑥)(𝜆(𝑠,𝑥,𝜍)−1)𝑓(𝑥)𝑑𝑥𝑁(𝜔,𝑑𝑠,𝑑𝜍). (4.22) Now, using integration by parts, we get _ℝ𝑢(𝑠,𝑥)ℒ_𝑠𝑓(𝑥)𝑑𝑥=_ℝ𝑓(𝑥)ℒ^∗_𝑠𝑢(𝑠,𝑥)𝑑𝑥. (4.23) Further it holds again integration by parts and by substitution that _ℝ𝑢(𝑠,𝑥)ℬ_𝑠𝑓(𝑥)𝑑𝑥=_ℝ𝑓(𝑥)ℬ^∗_𝑠𝑢(𝑠,𝑥)𝑑𝑥. (4.24) Fubini together with (4.23) and (4.24) then yields _ℝ𝑓(𝑥)𝑢(𝑡,𝑥)𝑑𝑥=_ℝ𝑓(𝑥)𝜉(𝑥)𝑑𝑥+_ℝ𝑓(𝑥)^𝑡₀𝒪^∗_𝑠𝑢()+𝑠,𝑥𝑑𝑠𝑑𝑥_ℝ𝑓(𝑥)^𝑡₀𝑢(𝑠,𝑥)ℎ(𝑠,𝑥)𝑑𝑊_𝑠(𝜔+)𝑑𝑥_ℝ𝑓(𝑥)^𝑡_0ℝ0𝑢(𝑠,𝑥)(𝜆(𝑠,𝑥,𝜍)−1)𝑁(𝜔,𝑑𝑠,𝑑𝜍)𝑑𝑥. (4.25) Since this holds for all 𝑓∈𝐶^∞₀(ℝ), we get (4.20).

Acknowledgments

This work was started when V. Mandrekar was at the CMA, University of Oslo. He thanks the CMA for support and Professor Øksendal for interesting discussions and hospitality.